zuko absolutely sees katara as the little sister azula could've been if not for their dad, but does not understand how to be a big brother in a way that isn't encouraging murder (see the southern raiders ep). katara refuses to acknowledge that she has a second big brother, but does admittedly appreciate having someone in her corner to encourage her more... destructive feelings
less than a month after zuko's been crowned firelord, the very day after katara's declared him healed, she marches into his war room (which someone really should rename, given that they're trying to end the war) and challenges him to an agni kai. she refuses to give any reason, just storms back out of the room.
the fire sages are all terrified and expecting zuko to redeclare war upon the southern water tribe. they're not expecting zuko to just sigh long-sufferingly and ask someone to make sure his schedule is cleared for sunset, because that's the most fair time for them to fight in terms of celestial boosts (zuko thinks this is normal sibling behaviour, no-one is going to correct him on this).
zuko's council realise that yes, he does intend on fighting Master Katara, the last southern waterbender, the very woman who defeated (at the time) Crown Princess Azula on the day that she was at her most powerful, and instead of having to undo all of their 'we're ending the war' decrees, they have to convince the firelord that fighting Master Katara, the last southern waterbender, the very woman who defeated (at the time) Crown Princess Azula on the day that she was at her most powerful is a terrible idea, particularly given that he was pissing blood a month ago.
zuko's not listening, cause obviously she isn't going to actually hurt him. probably. actually, she did once say that if he did anything she didn't like she would kill him- oh fuck has he somehow done something to piss katara off? oh well, only way to find out is to fight her
(the actual reason katara has challenged him to an agni kai is because she is bored shitless, and no-one is letting her into the meeting rooms where they're discussing reparations, which is stupid because she is LITERALLY THE LAST SOUTHERN WATER BENDER, she deserves to be reparated, and also a little bit cause she's mad at him for jumping in front of a lightning bolt that he definitely knew he could not redirect in a stupid attempt to save her life)
when reason clearly isn't working, the fire sages switch to telling zuko 'sorry bro you can't fight her, an agni kai is just between two firebenders’. this stops zuko for a good five minutes (it would’ve stopped him for a lot longer but he went to talk to azula about it (because she’s the person he knows who is best at agni kai’s, and she is also a scary, overpowered little sister, and because he wants to do right by azula and fix their bond) and she laughed at him and then told him 'you are the imposter that the peasantry is worshipping as their fire lord, just change the law')
zuko returns to the war room, where all of his council are trying to make contingency plans for if the scary little waterbending girl beats their last remaining candidate for firelord, and tells them 'new rule, anyone can challenge the fire lord to an agni kai', which is not at all what azula meant as is also an astronomically stupid idea.
an astronomically stupid idea that leads to katara challenging him to an agni kai whenever he pisses her off, which is at least twice for every visit she makes to the palace. (toph hears about this and joins in (she challenges him every time ty lee feels that the 'aura of the palace is misguided') which then obviously leads to sokka doing the same (whenever he's hungry). the fire sages are really mad about this because it's disrespecting and diminishing agni's will or whatever. zuko decides that he doesn't care, because it's changing the public atmosphere around agni kai's and they're becoming far less common due to the childish association with them)
after a year as fire lord, zuko is clinging to the fact that he managed to beat zhao. which is, objectively, a really sad and embarrassing thing to be proud of, but, oh well.
(the day that azula gets discharged, katara sends her a letter scheduling an agni kai with her for 'whenever you feel up for it. that is not a dig. i just don't want to fight you unless i'm fighting you at your best'. zuko is terrified by this prospect, azula is ecstatic. they both decide zuko is not allowed to be present and send him away to ba sing se or somewhere (the fire sages are deeply offended on zuko's behalf. he doesn't give a damn, because he knows that the two of them could absolutely force him out of the palace and he doesn't want to deal with that). he returns two days later to find that azula and katara are best friends and azula's given katara the secret stash of southern water bending scrolls that the fire nation kept hidden because 'it's not fair to fight you at a disadvantage either'.
zuko returns to ba sing se and spends a week trying to find a way to break to sokka that their little sisters are friends now and might be trying to overthrow him.)